We have seen in the previous page how the derivative is defined:
For a function *f*(*x*), its derivative at *x*=*a* is defined by

Let us give some examples.

**Example 1.** Let us start with the function
*f*(*x*) = *x*^{2}. We
have

So

which means

What about the derivative of
*f*(*x*) = *x*^{n}. Similar calculations,
using the binomial expansion for (*x*+*y*)^{n} (Pascal's Triangle),
yield

**Example 2.** Consider the function *f*(*x*)=1/*x* for .
We have

Consequently,

Have you noticed? The algebraic trick in both of the examples
above has been to factor out "*h*" in the numerator, so that we
can cancel it with the "*h*" in the denominator! This is what you
try to do whenever you are asked to compute a derivative using
the limit definition.

You may believe that every function has a derivative. Unfortunately that is not the case.

**Example 3.** Let us discuss the derivative of
*f*(*x*) = |*x*| at
0. We have

But

which implies that

**Remark.** This example is interesting. Even though the
derivative at the point does not exist, the right and the left
limit of the ratio do exist. In fact, if we use the
slope-interpretation of the derivative we see that this means
that the graph has two lines close to it at the point under
consideration. They could be seen as "half-tangents". See
Picture.

So let's push it a little bit more and ask whether a function always has a tangent or half-tangents at any point. That is not the case either.

**Example 4.** Let us consider the function
for ,
with
*f*(0) = 0. We have

Recall that the function has no limit when

What else can go wrong?

**Example 5.** Consider the function
.
Then
we have

Since

then

In fact, the way the concept of the tangent line was introduced is based on the notion of slope. You already know that vertical lines do not have slopes. So we say that the derivative does not exist whenever the tangent line is vertical. Nevertheless keep in mind that when the limit giving the derivative is then the function has a vertical tangent line at the point.

It can be quite laborious (or impossible) to compute the derivative by hand as we have done so far. In the next pages we will show how techniques of differentiation help bypass the limit calculations and make our life much easier.

**Exercise 1.** Find the derivative of

**Exercise 2.** Discuss the differentiability of

**Exercise 3.** We say that the graph of *f*(*x*) has a **cusp** at (*a*,*f*(*a*)), if *f*(*x*) is continuous at *a* and if the
following two conditions hold:

- 1.
- as from one side (left or right);
- 2.
- as from the other side.

Determine whether
*f*(*x*) = *x*^{4/3} and
*g*(*x*) = *x*^{3/5} have a
cusp at (0,0).

**Exercise 4.** Show that if *f* '(*a*) exists, then we have

**Exercise 5.** A spherical balloon is being inflated. Find
the rate at which its volume *V* is changing with respect to the
radius.

**
**

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Helmut Knaust

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